Subgroups — 3₋¹⁺²×#338 #13136017

Order 1728.

344 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×3₋¹⁺²	C₂	1	3	yes	maximal
2	C₃²×#338	C₃	1	1	yes	maximal, characteristic
3	C₉×#338	C₃	1	3	yes	maximal, complemented
4	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
5	M₁₆×3₋¹⁺²	—	2	6	no	—
6	C₂×C₈×3₋¹⁺²	C₄	1	6	yes	—
7	C₂×C₃²×M₁₆	C₂×C₃	1	3	yes	—
8	C₂×C₉×M₁₆	C₂×C₃	1	9	yes	—
9	C₂³×3₋¹⁺²	Q₈	1	1	yes	characteristic
10	C₂×C₄×3₋¹⁺²	C₂×C₄	1	3	yes	—
11	C₂×C₄×3₋¹⁺²	D₄	1	3	yes	—
12	C₈×3₋¹⁺²	—	2	6	no	—
13	C₃×#338	—	3	1	no	—
14	C₃×#338	C₃²	1	1	yes	characteristic
15	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
16	C₃²×M₁₆	—	2	6	no	—
17	C₂×C₃²×C₈	C₃×C₄	1	6	yes	—
18	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
19	C₉×M₁₆	—	2	18	no	—
20	C₂×C₈×C₉	C₃×C₄	1	18	yes	—
21	C₂²×3₋¹⁺²	—	4	1	no	—
22	C₂²×3₋¹⁺²	C₄⋊C₄	1	3	yes	—
23	C₄×3₋¹⁺²	C₄²	1	1	yes	characteristic
24	C₄×3₋¹⁺²	C₂²⋊C₄	1	3	yes	—
25	C₂×C₃×M₁₆	—	3	3	no	—
26	C₂×C₃×M₁₆	C₂×C₃²	1	3	yes	—
27	C₂³×C₃²	C₃×Q₈	1	1	yes	characteristic
28	C₂×C₃²×C₄	C₂×C₃×C₄	1	3	yes	—
29	C₂×C₃²×C₄	C₃×D₄	1	3	yes	—
30	C₂³×C₉	C₃×Q₈	1	3	yes	—
31	C₃²×C₈	—	2	6	no	—
32	C₂×C₄×C₉	C₂×C₃×C₄	1	9	yes	—
33	C₂×C₄×C₉	C₃×D₄	1	9	yes	—
34	C₈×C₉	—	2	18	no	—
35	#338	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
36	C₂×3₋¹⁺²	—	2	3	no	—
37	C₂×3₋¹⁺²	#95	1	1	yes	characteristic
38	C₂²×C₃×C₄	—	3	1	no	—
39	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
40	C₃×M₁₆	—	2	6	no	—
41	C₃×M₁₆	—	6	6	no	—
42	C₂×C₃×C₈	—	3	6	no	—
43	C₂×C₃×C₈	C₃²×C₄	1	6	yes	—
44	C₂²×C₃²	—	4	1	no	—
45	C₂²×C₃²	C₃×C₄⋊C₄	1	3	yes	—
46	C₃²×C₄	C₃×C₄²	1	1	yes	characteristic
47	C₃²×C₄	C₃×C₂²⋊C₄	1	3	yes	—
48	C₂²×C₉	—	4	3	no	—
49	C₂²×C₉	C₃×C₄⋊C₄	1	9	yes	—
50	C₄×C₉	C₃×C₄²	1	3	yes	—
51	C₄×C₉	C₃×C₂²⋊C₄	1	9	yes	—
52	C₂×M₁₆	C₂×3₋¹⁺²	1	3	yes	—
53	3₋¹⁺²	#338	1	1	yes	characteristic, direct product factor, Sylow
54	C₂³×C₃	—	3	1	no	—
55	C₂³×C₃	C₃²×Q₈	1	1	yes	characteristic
56	C₂×C₃×C₄	—	3	3	no	—
57	C₂×C₃×C₄	—	3	3	no	—
58	C₂×C₃×C₄	C₂×C₃²×C₄	1	3	yes	—
59	C₂×C₃×C₄	C₃²×D₄	1	3	yes	—
60	C₃×C₈	—	2	6	no	—
61	C₃×C₈	—	6	6	no	—
62	C₂×C₃²	—	2	3	no	—
63	C₂×C₃²	C₃×#95	1	1	yes	characteristic
64	C₂×C₉	—	2	9	no	—
65	C₂×C₉	C₃×#95	1	3	yes	—
66	C₂²×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
67	M₁₆	—	2	6	no	—
68	C₂×C₈	C₄×3₋¹⁺²	1	6	yes	—
69	C₂²×C₃	—	3	3	no	—
70	C₂²×C₃	—	4	1	no	—
71	C₂²×C₃	—	12	1	no	—
72	C₂²×C₃	C₃²×C₄⋊C₄	1	3	yes	—
73	C₃×C₄	—	3	1	no	—
74	C₃×C₄	—	3	3	no	—
75	C₃×C₄	C₃²×C₄²	1	1	yes	characteristic
76	C₃×C₄	C₃²×C₂²⋊C₄	1	3	yes	—
77	C₃²	C₃×#338	1	1	yes	characteristic
78	C₉	C₃×#338	1	3	yes	complemented
79	C₂³	Q₈×3₋¹⁺²	1	1	yes	characteristic
80	C₂×C₄	C₂×C₄×3₋¹⁺²	1	3	yes	—
81	C₂×C₄	D₄×3₋¹⁺²	1	3	yes	—
82	C₈	—	2	6	no	—
83	C₂×C₃	—	2	3	no	—
84	C₂×C₃	—	3	1	no	—
85	C₂×C₃	—	6	3	no	—
86	C₂×C₃	C₃²×#95	1	1	yes	characteristic
87	C₂²	—	4	1	no	—
88	C₂²	C₄⋊C₄×3₋¹⁺²	1	3	yes	—
89	C₄	C₄²×3₋¹⁺²	1	1	yes	characteristic
90	C₄	C₂²⋊C₄×3₋¹⁺²	1	3	yes	—
91	C₃	—	3	1	no	—
92	C₃	C₃²×#338	1	1	yes	characteristic
93	C₂	—	2	3	no	—
94	C₂	3₋¹⁺²×#95	1	1	yes	characteristic

	subgroup	G/H
center, commutator	C₃×C₄	C₃²×C₄²
Frattini, cosocle	C₂²×C₃×C₄	C₂²×C₃²
socle	C₂×C₃	C₃²×#95
prime radical	C₂³×C₃²	C₃×Q₈
LCS	G ▷ C₃×C₄ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₉×#338	C₃	semidirect
35. #338	3₋¹⁺²	direct (N × H)
53. 3₋¹⁺²	#338	direct (N × H)
78. C₉	C₃×#338	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×M₁₆×3₋¹⁺²	2
C₃²×#338	3
C₉×#338	3

Subgroups — 3−1+2×#338 #13136017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#338 #13136017